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G = C62.116C23order 288 = 25·32

111st non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.116C23, C628(C2×C4), C23.33S32, C6.70(S3×D4), (C2×Dic3)⋊12D6, (C22×C6).76D6, C2.5(Dic3⋊D6), C6.D411S3, C6.D1216C2, (C6×Dic3)⋊14C22, C223(C6.D6), (C2×C62).35C22, (C2×C6)⋊8(C4×S3), C6.39(S3×C2×C4), C32(S3×C22⋊C4), (C22×C3⋊S3)⋊5C4, (C2×C3⋊S3).62D4, C22.57(C2×S32), C328(C2×C22⋊C4), C3⋊S33(C22⋊C4), (C3×C6).162(C2×D4), (C23×C3⋊S3).2C2, (C2×C6.D6)⋊14C2, (C3×C6).71(C22×C4), C2.16(C2×C6.D6), (C3×C6.D4)⋊20C2, (C2×C6).135(C22×S3), (C22×C3⋊S3).77C22, (C2×C3⋊S3)⋊15(C2×C4), SmallGroup(288,622)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.116C23
C1C3C32C3×C6C62C6×Dic3C2×C6.D6 — C62.116C23
C32C3×C6 — C62.116C23
C1C22C23

Generators and relations for C62.116C23
 G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 1538 in 331 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×8], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×20], S3 [×20], C6 [×6], C6 [×11], C2×C4 [×8], C23, C23 [×10], C32, Dic3 [×4], C12 [×4], D6 [×64], C2×C6 [×6], C2×C6 [×11], C22⋊C4 [×4], C22×C4 [×2], C24, C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×34], C22×C6 [×2], C22×C6, C2×C22⋊C4, C3×Dic3 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], S3×C2×C4 [×4], S3×C23 [×3], C6.D6 [×4], C6×Dic3 [×4], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, S3×C22⋊C4 [×2], C6.D12 [×2], C3×C6.D4 [×2], C2×C6.D6 [×2], C23×C3⋊S3, C62.116C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×4], C22×S3 [×2], C2×C22⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×4], C6.D6 [×2], C2×S32, S3×C22⋊C4 [×2], C2×C6.D6, Dic3⋊D6 [×2], C62.116C23

Permutation representations of C62.116C23
On 24 points - transitive group 24T673
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 17 5 15 3 13)(2 18 6 16 4 14)(7 19 9 21 11 23)(8 20 10 22 12 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)

G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,19,9,21,11,23),(8,20,10,22,12,24)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,18,10,15),(8,17,11,14),(9,16,12,13)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)])

G:=TransitiveGroup(24,673);

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C4A···4H6A···6F6G···6Q12A···12H
order1222222222223334···46···66···612···12
size111122999918182246···62···24···412···12

48 irreducible representations

dim1111112222244444
type++++++++++++++
imageC1C2C2C2C2C4S3D4D6D6C4×S3S32S3×D4C6.D6C2×S32Dic3⋊D6
kernelC62.116C23C6.D12C3×C6.D4C2×C6.D6C23×C3⋊S3C22×C3⋊S3C6.D4C2×C3⋊S3C2×Dic3C22×C6C2×C6C23C6C22C22C2
# reps1222182442814214

Matrix representation of C62.116C23 in GL6(𝔽13)

1120000
100000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
0000121
0000120
,
800000
080000
0011000
0051200
000001
000010
,
050000
500000
0012300
008100
000010
000001
,
1200000
0120000
001000
0051200
000010
000001

G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,5,0,0,0,0,10,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C62.116C23 in GAP, Magma, Sage, TeX

C_6^2._{116}C_2^3
% in TeX

G:=Group("C6^2.116C2^3");
// GroupNames label

G:=SmallGroup(288,622);
// by ID

G=gap.SmallGroup(288,622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=b^3*d>;
// generators/relations

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